Trajectory Modeling by Shape
Trajectory Modeling by Shape Nicholas P. Jewell Departments of Statistics & School of Public Health (Biostatistics) University of California, Berkeley San Diego Chapter American Statistical Association Pfizer, La Jolla October 16, 2013 1
Thanks • Joint work with Brianna Heggeseth (Williams College)
References • Heggeseth, BC and Jewell, NP. The impact of covariance misspecification in multivariate Gaussian mixtures on estimation and inference: an application to longitudinal modeling. Statistics in Medicine, 2013, 32, 2790-2803 . • Heggeseth, BC and Jewell, NP. Clustering trajectories by shape. To appear.
“Understanding our world requires conceptualizing the similarities and differences between the entities that compose it” Robert Tryon and Daniel Bailey, 1970
3
How does BMI change with age?
National Longitudinal Study of Youth (NLSY) from 1979 - 2008. 4
How does BMI change with age?
National Longitudinal Study of Youth (NLSY) from 1979 - 2008. 5
How does BMI change with age?
National Longitudinal Study of Youth (NLSY) from 1979 - 2008. 6
Typical Longitudinal Analysis • Use Generalized Estimating Equations (GEE) to estimate the mean outcome, and how it changes over time, adjusting for covariates regression parameter estimation is consistent despite potential covariance misspecification efficiency can be gained through use of a more appropriate working correlation structure robust (sandwich) standard error estimators available • But, with a heterogeneous population, BMI does not change much for some people as they age BMI changes considerably for some people as they age • We don’t wish to average out these separate trajectories by modeling the mean over time 7
Finite Mixture Models • Data for n individuals:
yi = (yi1 , . . . , yim ) measured at times
ti = (ti1 , . . . , timi ) • We assume K latent trajectories in the population that are distributed with frequencies: ⇡1 , . . . , ⇡K
f (y|t, ✓) = ⇡1 f (y|t,
where ⇡k > 0 and
1 , ⌃1 )
⌃K k=1 ⇡k = 1 .
+ · · · + ⇡K f (y|t,
K , ⌃K )
f (y|t, k , ⌃k ) , a multivariate and covariance ⌃k .
• The (conditional) mixture density is Gaussian with mean
µk
• In most trajectory software, (conditional) independence is assumed as a working correlations structure:
(⌃k =
✓ = (⇡1 , . . . , ⇡K ;
2 k I). 1, . . . ,
K ; ⌃1 , . . . , ⌃K )
8
Finite Mixture Models • The mean vector Linear: Quadratic:
µk
is related to the observation times as follows:
(µk )j =
0
(µk )j =
+
1 tij
0+
1 tij +
2 t 2 ij
Splines in observation times where the regression model (and coefficients) are assumed the same for each cluster, and tij is the jth observation for the ith individual where 1 j mi
9
Finite Mixture Models • Group membership:
exp( k z) ⇡k = K ⌃j=1 exp( j z)
Z is set of same or different covariates This expands ✓ to include the
s also
10
Estimation for Mixture Models • Maximum likelihood estimation for θ via the EM algorithm
• K is pre-specified; can be chosen using the BIC • Parameter estimators are not consistent under covariance misspecification (White, 1982; Heggeseth and Jewell, 2013). • Robust (sandwich) standard error estimators are available. • How bad can the bias in regression estimators be? What influences its size?
11
Mispecified Covariance Structure Bias and Separation of Trajectories • Separated components lead to little bias even when you wrongly assume independence.
Black dashed -- true means, Solid lines – estimated means ˆ I ( 01 ) = 0.01, SE ˆ R ( 01 ) = 0.01 12 ˆ I ( 01 ) = 0.02, SE ˆ R ( 01 ) = 0.06 SE SE
Mispecified Covariance Structure Bias and Level of Dependence • Components with little dependence lead to small bias even when you wrongly assume independence.
Black dashed -- true means, Solid lines – estimated means
ˆ I( SE
01 )
ˆ R( = 0.02, SE
01 )
= 0.06
ˆ I( SE
01 )
ˆ R( = 0.03, SE
01 )
= 0.04
13
NLSY Data Analysis
Covariance makes a difference to the trajectories hard to estimate bias from mispecified covariance
14
How Do We Group These Blocks?
15
Group by Color
16
Group by Shape
17
How Do We Group These Blocks?
18
Group by Color or Shape
19
How Do We Group These (Regression) Lines?
20
y
15
10
5
2
4
6 x
8
10
20
Group by Intercept
20
y
15
10
5
2
4
6
8
10
x
21
Group by Level
20
y
15
10
5
2
4
6
8
10
x
22
Group by Shape (Slope)
20
y
15
10
5
2
4
6 x
8
10
23
Simulated Data
Y
15
10
5
0 2
4
6
8
10
Time
How could we group these individuals?
24
Simulated Data
15
Y
●
10
● ●
● ●
5
0 2
4
6
8
10
Time
How could we group these individuals?
25
Real Longitudinal Data • Center for the Health Assessment of Assessment of Mothers and Children of Salinas (CHAMACOS) Study In 1999-2000, enrolled 601 pregnant women in agricultural Salinas Valley, CA. Mostly Hispanic, agricultural workers. Determine if exposure to pesticides and other chemicals impact children's growth patterns (BMI, neurological measures etc_. • First, focus on studying/estimating the growth patterns of children. • Second, determine if early life predictors are related to the patterns pesticide exposure BPA (bisphenol A)
26
CHAMACOS Data 40
35
BMI
30
25
20
15
10 20
40
60
80
100
120
Age in months
How could we group these individuals?
27
Cluster Analyses • Clustering is the task of assigning a set of objects into groups so that the objects in the same group are more similar to each other than to those in other groups. • What does it mean for objects to be more similar or more dissimilar? Distance matrix • Why do we cluster objects?
28
Standard Clustering Methods • Partition methods Partition objects into K groups so that an objective function of dissimilarities is minimized or maximized. Example: K-means Algorithm • Model-based methods Assume a model that includes a grouping structure and estimate parameters. Example: Finite Mixture Models
29
K-means algorithm • Input: Data for n individuals in vector form. For individual i , the observed data vector is
yi = (y1i , . . . , yim ). • Measure of Dissimilarity: Squared Euclidean distance. The dissimilarity between the 1st and 2nd individuals is
d(y1
y2 ) = ky1
y2 k2 = (y11
y12 )2 + · · · + (yim
y2m )2
30
K-means Algorithm • Goal: Partition individuals into K sets
C = {C1 , C2 , . . . , CK } so as
to minimize the within-cluster sum of squares
⌃K k=1 ⌃yi 2Ck kyi where
µk
µk k2
is the mean vector of individuals in Ck.
(K must be known before starting K-means. There are many ways to choose K from the data that try to minimize the dissimilarity within each cluster while maximizing the dissimilarity between clusters: for example, the use of silhouettes.) 31
Application to Simulated Data K−means
Y
15
10
5
0 2
4
6 Time
8
10
32
Application to Simulated Data K−means
Y
15
●
●
●
●
●
●
●
●
●
●
10
5
0 2
4
6
8
10
Time
How would you describe—interpret—the group trajectories?
33
Finite Mixture Model Applied to CHAMACOS Data Mixture Model with Independence 40
35
BMI
30
25
20
15
10 20
40
60
80
Age in months
100
120
34
Finite Mixture Model Applied to CHAMACOS Data Mixture Model with Independence 40
35
BMI
30
25
20
15
10 20
40
60
80
Age in months
100
120
35
Finite Mixture Model Applied to CHAMACOS Data Mixture Model with Exponential 40
35
BMI
30
25
20
15
10 20
40
60
80
Age in months
100
120
36
Clustering by Shape • Interested in shape not just level (which appears to dominate clustering techniques) •
Want a method that: Works with irregularly sampled data Includes a way to estimate the relationship between baseline risk factors and group membership Groups individuals according to the outcome pattern over time ignoring the level
37
Clustering by Shape Options • Estimate slopes between neighboring observations and cluster on the “derived” observations • Fit splines for each individual, differentiate, and cluster on coefficients of resulting derivative • Use partition based cluster methods (like PAM) but use (i) the Pearson coefficient as a distance or dissimilarity measure
dcorr (x, y) = 1
Corr(x, y)
or the cosine-angle measure of dissimilarity
dcos (x, y) = 1
⌃m j=1 xj yj 2 m 2 (⌃m j=1 xj )(⌃j=1 yj )
• Vertical shifting individual trajectories 38
Vertical Shifting • For each individual, calculate
yi⇤
= yi
mi
1
mi ⌃j=1 yij
• Each individual now has mean zero and so level is removed from any resulting clustering • Apply clustering technique to shifted data, e.g. finite mixture model
39
Correlation Models for Vertical Shifted Data • Without specifying group, suppose
⇤ yi
=
where
i Imi
Imi
is an
+ µi + ✏i ,
⇠ F , ✏ ⇠ N (), ⌃)
mi length vector of 1s, and
µij = µk (tij ) is the j
th
element of the vector of mean values for
the kth group evaluated at the observation times ti . Thus,
⇤ yi
= Ai yi = µi
µ¯i + ✏i
✏¯i 40
Correlation Models for Vertical Shifted Data ⇤ Cov(Yi
µi ) = Cov((A
Imi )µi + A✏)
Two components of the covariance One induced by the averaging process One induced by (random) observation times
41
Correlation Models for Vertical Shifted Data Observation Times Fixed
Cov(Y⇤
µ) = A⌃AT
suppressing the individual/group indices for simplicity (Σ is allowed to vary across clusters)
This covariance matrix is singular since
det(A) = 0
This naturally reflects the “loss” of one dimension 42
Correlation Models for Vertical Shifted Data Observation Times Fixed
Cov(Y⇤ • If
⌃=
2
I
µ) = A⌃AT
(conditional independence with constant variance,
then the induced covariance is exchangeable with negative correlation given by
1/(m
1) and variance decreases to
2
m 1 ) m
• If original covariance is exchangeable with constant variance and correlation ρ then the induced covariance remains exchangeable with negative correlation and reduced variance
2
(1
⇢)
m 1 ) m
43
Correlation Models for Vertical Shifted Data Observation Times Fixed
Cov(Y⇤ If
⌃=
2
I
µ) = A⌃AT
(conditional independence with constant variance,
then the induced covariance is exchangeable with negative correlation m 1 given by 1/(m 1) and variance decreases to 2 m ) This induced exchangeable correlation is the lower bound for correlation in an exchangeable matrix Thus, if “estimated”, the (true) parameter is on the boundary of the parameter space
44
Correlation Models for Vertical Shifted Data Observation Times Random (µ is random) Cov(Y⇤
µ) = m
2
0 2 T T ⌃m V ar(t )[µ (E(t ))] 11 + A⌃A j j j=1
Sum of two non-invertible matrices, but the positive magnitude of the first matrix may counteract the negative correlations of the second.
45
Correlation Models for Vertical Shifted Data Observation Times Random (µ is random) Cov(Y⇤
µ) = m
500 simulations of
2
0 2 T T ⌃m V ar(t )[µ (E(t ))] 11 + A⌃A j j j=1
Yi = µi + ✏i
✏i ⇠ N (0, ⌃⇢ )
where the error covariance matrix is of exponential form with range ρ
t=T+⌧
⌧ ⇠ N (0,
2 ⌧ I)
T = (1, 2, . . . , 9, 10)
µij = µ(tij ) 46
Correlation Models for Vertical Shifted Data
47
Simulated Data
Y
15
10
5
0 2
4
6
8
10
Time
How could we group these individuals?
48
Application to Simulated Data K−means
Y
15
●
●
●
●
●
●
●
●
●
●
10
5
0 2
4
6
8
10
Time
How would you describe—interpret—the group trajectories?
49
Vertical Shifting Applied to Simulated Data Vertically Shifted Mixture Model with Exponential
Y
15
10
5
0 2
4
6 Time
8
10
50
Vertical Shifting Applied to Simulated Data Vertically Shifted Mixture Model with Exponential
Y
15
10
5
0 2
4
6 Time
8
10
51
500 Simulations
µ1 (t) =
1
t
µ2 (t) = 11
t
µ3 (t) = 0 µ4 (t) =
negative slope, low level negative slope, high level zero slope middle level
11 + t
µ5 (t) = 1 + t
positive slope, low level
positive slope, high level
Mean functions evaluated at five equidistant points that span [1,10} Including ends of the interval 52
500 Simulations
µ1 (t) =
1
t
µ2 (t) = 11
t
µ3 (t) = 0 µ4 (t) =
negative slope, low level negative slope, high level zero slope middle level
11 + t
µ5 (t) = 1 + t
positive slope, low level
positive slope, high level
Two components to noise: random individual level perturbationN (0, 2 ) random measurement error across times (exchangeable correlation)
N (0, 53
2 ✏)
500 Simulations
54
Vertical Shifting with CHAMACOS Vertically Shifted Mixture Model with Exponential 40
35
BMI
30
25
20
15
10 20
40
60
80
100
120
Age in months
55
Further Thoughts • Interest often focuses on the role of explanatory covariates as well as the trajectories themselves: two-part models: (i) model for overall level (ii) model for shape trajectory
• Time-dependent covariates
56
Thanks • Joint work with Brianna Heggeseth (Williams College)
References • Heggeseth, BC and Jewell, NP. The impact of covariance misspecification in multivariate Gaussian mixtures on estimation and inference: an application to longitudinal modeling. Statistics in Medicine, 2013, 32, 2790-2803 . • Heggeseth, BC and Jewell, NP. Clustering trajectories by shape. To appear.
“Understanding our world requires conceptualizing the similarities and differences between the entities that compose it” Robert Tryon and Daniel Bailey, 1970
3
How does BMI change with age?
National Longitudinal Study of Youth (NLSY) from 1979 - 2008. 4
How does BMI change with age?
National Longitudinal Study of Youth (NLSY) from 1979 - 2008. 5
How does BMI change with age?
National Longitudinal Study of Youth (NLSY) from 1979 - 2008. 6
Typical Longitudinal Analysis • Use Generalized Estimating Equations (GEE) to estimate the mean outcome, and how it changes over time, adjusting for covariates regression parameter estimation is consistent despite potential covariance misspecification efficiency can be gained through use of a more appropriate working correlation structure robust (sandwich) standard error estimators available • But, with a heterogeneous population, BMI does not change much for some people as they age BMI changes considerably for some people as they age • We don’t wish to average out these separate trajectories by modeling the mean over time 7
Finite Mixture Models • Data for n individuals:
yi = (yi1 , . . . , yim ) measured at times
ti = (ti1 , . . . , timi ) • We assume K latent trajectories in the population that are distributed with frequencies: ⇡1 , . . . , ⇡K
f (y|t, ✓) = ⇡1 f (y|t,
where ⇡k > 0 and
1 , ⌃1 )
⌃K k=1 ⇡k = 1 .
+ · · · + ⇡K f (y|t,
K , ⌃K )
f (y|t, k , ⌃k ) , a multivariate and covariance ⌃k .
• The (conditional) mixture density is Gaussian with mean
µk
• In most trajectory software, (conditional) independence is assumed as a working correlations structure:
(⌃k =
✓ = (⇡1 , . . . , ⇡K ;
2 k I). 1, . . . ,
K ; ⌃1 , . . . , ⌃K )
8
Finite Mixture Models • The mean vector Linear: Quadratic:
µk
is related to the observation times as follows:
(µk )j =
0
(µk )j =
+
1 tij
0+
1 tij +
2 t 2 ij
Splines in observation times where the regression model (and coefficients) are assumed the same for each cluster, and tij is the jth observation for the ith individual where 1 j mi
9
Finite Mixture Models • Group membership:
exp( k z) ⇡k = K ⌃j=1 exp( j z)
Z is set of same or different covariates This expands ✓ to include the
s also
10
Estimation for Mixture Models • Maximum likelihood estimation for θ via the EM algorithm
• K is pre-specified; can be chosen using the BIC • Parameter estimators are not consistent under covariance misspecification (White, 1982; Heggeseth and Jewell, 2013). • Robust (sandwich) standard error estimators are available. • How bad can the bias in regression estimators be? What influences its size?
11
Mispecified Covariance Structure Bias and Separation of Trajectories • Separated components lead to little bias even when you wrongly assume independence.
Black dashed -- true means, Solid lines – estimated means ˆ I ( 01 ) = 0.01, SE ˆ R ( 01 ) = 0.01 12 ˆ I ( 01 ) = 0.02, SE ˆ R ( 01 ) = 0.06 SE SE
Mispecified Covariance Structure Bias and Level of Dependence • Components with little dependence lead to small bias even when you wrongly assume independence.
Black dashed -- true means, Solid lines – estimated means
ˆ I( SE
01 )
ˆ R( = 0.02, SE
01 )
= 0.06
ˆ I( SE
01 )
ˆ R( = 0.03, SE
01 )
= 0.04
13
NLSY Data Analysis
Covariance makes a difference to the trajectories hard to estimate bias from mispecified covariance
14
How Do We Group These Blocks?
15
Group by Color
16
Group by Shape
17
How Do We Group These Blocks?
18
Group by Color or Shape
19
How Do We Group These (Regression) Lines?
20
y
15
10
5
2
4
6 x
8
10
20
Group by Intercept
20
y
15
10
5
2
4
6
8
10
x
21
Group by Level
20
y
15
10
5
2
4
6
8
10
x
22
Group by Shape (Slope)
20
y
15
10
5
2
4
6 x
8
10
23
Simulated Data
Y
15
10
5
0 2
4
6
8
10
Time
How could we group these individuals?
24
Simulated Data
15
Y
●
10
● ●
● ●
5
0 2
4
6
8
10
Time
How could we group these individuals?
25
Real Longitudinal Data • Center for the Health Assessment of Assessment of Mothers and Children of Salinas (CHAMACOS) Study In 1999-2000, enrolled 601 pregnant women in agricultural Salinas Valley, CA. Mostly Hispanic, agricultural workers. Determine if exposure to pesticides and other chemicals impact children's growth patterns (BMI, neurological measures etc_. • First, focus on studying/estimating the growth patterns of children. • Second, determine if early life predictors are related to the patterns pesticide exposure BPA (bisphenol A)
26
CHAMACOS Data 40
35
BMI
30
25
20
15
10 20
40
60
80
100
120
Age in months
How could we group these individuals?
27
Cluster Analyses • Clustering is the task of assigning a set of objects into groups so that the objects in the same group are more similar to each other than to those in other groups. • What does it mean for objects to be more similar or more dissimilar? Distance matrix • Why do we cluster objects?
28
Standard Clustering Methods • Partition methods Partition objects into K groups so that an objective function of dissimilarities is minimized or maximized. Example: K-means Algorithm • Model-based methods Assume a model that includes a grouping structure and estimate parameters. Example: Finite Mixture Models
29
K-means algorithm • Input: Data for n individuals in vector form. For individual i , the observed data vector is
yi = (y1i , . . . , yim ). • Measure of Dissimilarity: Squared Euclidean distance. The dissimilarity between the 1st and 2nd individuals is
d(y1
y2 ) = ky1
y2 k2 = (y11
y12 )2 + · · · + (yim
y2m )2
30
K-means Algorithm • Goal: Partition individuals into K sets
C = {C1 , C2 , . . . , CK } so as
to minimize the within-cluster sum of squares
⌃K k=1 ⌃yi 2Ck kyi where
µk
µk k2
is the mean vector of individuals in Ck.
(K must be known before starting K-means. There are many ways to choose K from the data that try to minimize the dissimilarity within each cluster while maximizing the dissimilarity between clusters: for example, the use of silhouettes.) 31
Application to Simulated Data K−means
Y
15
10
5
0 2
4
6 Time
8
10
32
Application to Simulated Data K−means
Y
15
●
●
●
●
●
●
●
●
●
●
10
5
0 2
4
6
8
10
Time
How would you describe—interpret—the group trajectories?
33
Finite Mixture Model Applied to CHAMACOS Data Mixture Model with Independence 40
35
BMI
30
25
20
15
10 20
40
60
80
Age in months
100
120
34
Finite Mixture Model Applied to CHAMACOS Data Mixture Model with Independence 40
35
BMI
30
25
20
15
10 20
40
60
80
Age in months
100
120
35
Finite Mixture Model Applied to CHAMACOS Data Mixture Model with Exponential 40
35
BMI
30
25
20
15
10 20
40
60
80
Age in months
100
120
36
Clustering by Shape • Interested in shape not just level (which appears to dominate clustering techniques) •
Want a method that: Works with irregularly sampled data Includes a way to estimate the relationship between baseline risk factors and group membership Groups individuals according to the outcome pattern over time ignoring the level
37
Clustering by Shape Options • Estimate slopes between neighboring observations and cluster on the “derived” observations • Fit splines for each individual, differentiate, and cluster on coefficients of resulting derivative • Use partition based cluster methods (like PAM) but use (i) the Pearson coefficient as a distance or dissimilarity measure
dcorr (x, y) = 1
Corr(x, y)
or the cosine-angle measure of dissimilarity
dcos (x, y) = 1
⌃m j=1 xj yj 2 m 2 (⌃m j=1 xj )(⌃j=1 yj )
• Vertical shifting individual trajectories 38
Vertical Shifting • For each individual, calculate
yi⇤
= yi
mi
1
mi ⌃j=1 yij
• Each individual now has mean zero and so level is removed from any resulting clustering • Apply clustering technique to shifted data, e.g. finite mixture model
39
Correlation Models for Vertical Shifted Data • Without specifying group, suppose
⇤ yi
=
where
i Imi
Imi
is an
+ µi + ✏i ,
⇠ F , ✏ ⇠ N (), ⌃)
mi length vector of 1s, and
µij = µk (tij ) is the j
th
element of the vector of mean values for
the kth group evaluated at the observation times ti . Thus,
⇤ yi
= Ai yi = µi
µ¯i + ✏i
✏¯i 40
Correlation Models for Vertical Shifted Data ⇤ Cov(Yi
µi ) = Cov((A
Imi )µi + A✏)
Two components of the covariance One induced by the averaging process One induced by (random) observation times
41
Correlation Models for Vertical Shifted Data Observation Times Fixed
Cov(Y⇤
µ) = A⌃AT
suppressing the individual/group indices for simplicity (Σ is allowed to vary across clusters)
This covariance matrix is singular since
det(A) = 0
This naturally reflects the “loss” of one dimension 42
Correlation Models for Vertical Shifted Data Observation Times Fixed
Cov(Y⇤ • If
⌃=
2
I
µ) = A⌃AT
(conditional independence with constant variance,
then the induced covariance is exchangeable with negative correlation given by
1/(m
1) and variance decreases to
2
m 1 ) m
• If original covariance is exchangeable with constant variance and correlation ρ then the induced covariance remains exchangeable with negative correlation and reduced variance
2
(1
⇢)
m 1 ) m
43
Correlation Models for Vertical Shifted Data Observation Times Fixed
Cov(Y⇤ If
⌃=
2
I
µ) = A⌃AT
(conditional independence with constant variance,
then the induced covariance is exchangeable with negative correlation m 1 given by 1/(m 1) and variance decreases to 2 m ) This induced exchangeable correlation is the lower bound for correlation in an exchangeable matrix Thus, if “estimated”, the (true) parameter is on the boundary of the parameter space
44
Correlation Models for Vertical Shifted Data Observation Times Random (µ is random) Cov(Y⇤
µ) = m
2
0 2 T T ⌃m V ar(t )[µ (E(t ))] 11 + A⌃A j j j=1
Sum of two non-invertible matrices, but the positive magnitude of the first matrix may counteract the negative correlations of the second.
45
Correlation Models for Vertical Shifted Data Observation Times Random (µ is random) Cov(Y⇤
µ) = m
500 simulations of
2
0 2 T T ⌃m V ar(t )[µ (E(t ))] 11 + A⌃A j j j=1
Yi = µi + ✏i
✏i ⇠ N (0, ⌃⇢ )
where the error covariance matrix is of exponential form with range ρ
t=T+⌧
⌧ ⇠ N (0,
2 ⌧ I)
T = (1, 2, . . . , 9, 10)
µij = µ(tij ) 46
Correlation Models for Vertical Shifted Data
47
Simulated Data
Y
15
10
5
0 2
4
6
8
10
Time
How could we group these individuals?
48
Application to Simulated Data K−means
Y
15
●
●
●
●
●
●
●
●
●
●
10
5
0 2
4
6
8
10
Time
How would you describe—interpret—the group trajectories?
49
Vertical Shifting Applied to Simulated Data Vertically Shifted Mixture Model with Exponential
Y
15
10
5
0 2
4
6 Time
8
10
50
Vertical Shifting Applied to Simulated Data Vertically Shifted Mixture Model with Exponential
Y
15
10
5
0 2
4
6 Time
8
10
51
500 Simulations
µ1 (t) =
1
t
µ2 (t) = 11
t
µ3 (t) = 0 µ4 (t) =
negative slope, low level negative slope, high level zero slope middle level
11 + t
µ5 (t) = 1 + t
positive slope, low level
positive slope, high level
Mean functions evaluated at five equidistant points that span [1,10} Including ends of the interval 52
500 Simulations
µ1 (t) =
1
t
µ2 (t) = 11
t
µ3 (t) = 0 µ4 (t) =
negative slope, low level negative slope, high level zero slope middle level
11 + t
µ5 (t) = 1 + t
positive slope, low level
positive slope, high level
Two components to noise: random individual level perturbationN (0, 2 ) random measurement error across times (exchangeable correlation)
N (0, 53
2 ✏)
500 Simulations
54
Vertical Shifting with CHAMACOS Vertically Shifted Mixture Model with Exponential 40
35
BMI
30
25
20
15
10 20
40
60
80
100
120
Age in months
55
Further Thoughts • Interest often focuses on the role of explanatory covariates as well as the trajectories themselves: two-part models: (i) model for overall level (ii) model for shape trajectory
• Time-dependent covariates
56
